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Addition and Subtract

To add or subtract numbers using the rules for significant digits, you need to keep this idea in mind: you can't add or subtract from something you do not know.

The best method for keeping track of significant digits in addition and subtraction questions is to write the problem down as if you were going to do it by hand (though you may well use a calculator to do the actual math). Make sure the numbers are aligned at the decimal point. Add and subtract to get the full answer, then round it off at the farthest left significant digit in the answer.shown in the above examples, it is usually a good idea to carry out the calculation as if significant figures did not exist, then round off the answer to the farthest left significant digit at the end.

Significant Digits

Since significant figures are a method of keeping track of the amount of error in experimental results, they only apply to experimental measurements. Some numbers are exact, and are not affected by the number of significant digits.

Plus Minus Notation

Plus Minus Notation
An even better way to show the amount of uncertainty in a number is to use notation. When you do this there is no question about how many significant digits there are in a measurement.

significant digits

What we are doing here is using the concept of significant digits. Think about the original ruler. We record all the numbers of which we are certain (the 9.6) and the first digit we are not sure of (the 0.00). The number must be recorded as 9.60. Now just count the digits in this number. There are three, so 9.60 has 3 significant digits.

Recording results in significant numbers

Record your results with all the digits that you can measure, to the limit of the uncertainty of the equipment. Make sure to include zeroes when they are actually being measured.

Calculations with Significant Digits

The first critical step in doing calculations with significant digits is to make sure that the data you are using is correctly recorded. Since significant figures are a method of keeping track of the amount of error in experimental results, they only apply to experimental measurements. Some numbers are exact, and are not affected by the number of significant digits.

Every result of a calculation that involves experimental results -- whether addition, subtraction, multiplication or division -- must report its results correctly using significant digits.

Exact numbers

Only experimentally measured numbers or results calculated from them use significant digits.

Not every number is measured experimentally. Some numbers are exact. They have an infinite number of significant digits. We usually consider the following kinds of numbers to be exact:

When you use an exact number in a calculation, it has no effect on the number of significant digits you report in your final answer.

The number of Significant Digits

Significant digits are all the numbers that are certain, and one digit that contains some uncertainty.

When you record a number from an experiment, you must make sure that you write down the digits correctly. When doing calculations, we need to know the number of significant digits. In most numbers this is easy. Just count the digits (not the number of decimal places) in the number.

Exact mearsure is impossile?

It is impossible to make an exact measurement.Therefore, all experimental results are wrong. Just how wrong they are depends on the kinds of errors that were made in the experiment.

How do you minimize error?

However, one of their goals is to minimize errors, and to be aware of what the errors may be. Significant digits is one way of keeping track of how much error there is in a measurement

HOw do you record your results?

They are far more likely to say: "it is likely that ..." or "it is probable that ..." than to give an exact answer.

As a science student you too must be careful to learn how good your results are, and to report them in a way that indicates your confidence in your answers.

Random Error

Random Errors
These errors are unpredictable. They are chance variations in the measurements over which you as experimenter have little or no control. There is just as great a chance that the measurement is too big as that it is too small.

Since the errors are equally likely to be high as low, averaging a sufficiently large number of results will, in principle, reduce their effect

Systematic Errors

Systematic Errors
These are errors caused by the way in which the experiment was conducted. In other words, they are caused by the design of the system.

Systematic errors can not be eliminated by averaging In principle, they can always be eliminated by changing the way in which the experiment was done. In actual fact though, you may not even know that the error exists.

Why is it not easy to discuss the idea of Systematic errors?

It is not easy to discuss the idea of systematic and random errors without referring to the procedure of an experiment. Here is a procedure for a simple experiment to measure the density of rubbing alcohol (iso-propanol).

Mean

Normal Distribution When data is normally distributed its shape is a bell curve. The mean is the centre point of the curve, and half the results are higher than the mean, half lower.
Mean The measure of central tendancy we traditionally call the "average". To find the mean, sum all the results, and divide by the number of results.

Human Error

What students seem to mean by human errors are really mistakes. Spilling part of a solution, dropping part of a solid from the weighing paper, or doing a calculation wrong are blunders, not errors. They can be avoided by being careful. If you know that you have made such a mistake  a "human" error  you simply cannot use the results. You must discard the measurements if you know that these kinds of mistakes have happened and redo the observations, or redo the calculations properly.

WHy don't you record human error?

Never report these things as "human error". They are mistakes that should not have happened. spilling, or sloppiness, dropping the equiment, etc.
bad calculations, doing math incorrectly, or using the wrong formula
reading a measuring device incorrectly (thermometer, balance, etc.)
not cleaning the equipment
using the wrong chemical
not following the planned procedure

Recording Measuements

If you are going to understand significant digits, you have to be sure you are recording your measurements properly. Sounds simple, right? And it is, yet it also has a few "gotcha's" to be careful about.

Accuracy

Accuracy indicates how close a measurement is to the accepted value. For example, we'd expect a balance to read 100 grams if we placed a standard 100 g weight on the balance. If it does not, then the balance is inaccurate.

Precision

Precision indicates how close together or how repeatable the results are. A precise measuring instrument will give very nearly the same result each time it is used.
There are several ways to report the precision of results. The simplest is the range (the difference between the highest and lowest results) often reported as a deviation from the average.

Adding and subtract round

The best method for keeping track of significant digits in addition and subtraction questions is to write the problem down as if you were going to do it by hand (though you may well use a calculator to do the actual math). Make sure the numbers are aligned at the decimal point. Add and subtract to get the full answer, then round it off at the farthest left significant digit in the answer

multiplying Dividing

Round the answer to the shortest number of significant digits in the numbers you are multiplying or dividing.It is usually a good idea to carry out the calculation as if significant figures did not exist, then round off the answer to the proper significant digits at the end.

Scientific Notation

Always write numbers in scientific notation if there is any confusion about the number of significant digits they contain.
There is no difference in dealing with numbers written in scientific notation. In fact, its easier, because there is never any confusion over whether the zeroes are significant or not. Just count the number of digits in the shortest experimental number, multiply (or divide) the numbers, and then round off the answer at the shortest number of digits.